Casimir operator in su(2)

hjlim

Hi all,

If I define T_ij = a⁺_i a_j, then

C₂ = T₁₁T₁₁ + T₁₂T₂₁ + T₂₁T₁₂ + T₂₂T₂₂ is a second order casimir operator.

For SU(2), it's [tex]\frac{N}{2}[/tex] ([tex]\frac{N}{2}[/tex] + 1)

But as I calculate it directly,
C₂ = a⁺₁ a₁a⁺₁ a₁ + a⁺₁ a₂a⁺₂ a₁ + a⁺₂ a₁a⁺₁ a₁ + a⁺₂ a₂a⁺₂ a₂ =
a⁺₁ a₁a⁺₁ a₁ + a⁺₁ (a⁺₂a₂ + 1)a₁ + a⁺₂ (a⁺₁a₁ + 1)a₂ + a⁺₂ a₂a⁺₂ a₂ =
N₁N₁ + N₁(N₂ + 1) + N₂(N₁ + 1) + N₂N₂ = (N₁ + N₂)² + N₁ + N₂ = N(N + 1)

which is different from above. Can you let me know what is wrong with my argument? Thank you very much!